Exercise 23. Let φ(z) = z/(1-Iz) for all E (-1,1). (a) Show that p is a...
(k)) In the power method, let r,-φ(z(k+1))/φ(z(k)). We know that limk-oork Show that the relative errors obey We know Ai. (는) Ck where the numbers ck form a convergent (and hence bounded) sequence. (k)) In the power method, let r,-φ(z(k+1))/φ(z(k)). We know that limk-oork Show that the relative errors obey We know Ai. (는) Ck where the numbers ck form a convergent (and hence bounded) sequence.
4.(10pts) Write Laplaces' equation in cylindricaol co-ordinates(p527 ex.3,use pinstead ofr) Assume the solution, e, φ, z), n can be written φ (p, φ, z)s u(p, φ)e-kz and Show that the equation for u is the two dimensional wave equation; Written in polar co-ordinates:xpcosp,y psinp For a plane wave traveling in a direction defined by:4-kcosce, ky-kinα Show that the plane wave solution can be written; look for a solution u z(x)en (2-n212,-0 And the equation for Z, is Bessels equation:Zh "x2...
7. Let V = Pa(R), the vector space of polynomials over R of degree less than 2, with inner product Define φ E p by φ(g)-g(-1) a) By direct calculation, find f e V such that (S)-dg). You are given that A 1, V3-2v) is an orthonormal basis for V (you do not need to check this). b) Find the same f as in part a, using the formula for A(6) from class. 7. Let V = Pa(R), the vector...
Please explain Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute V(X) (Hint: use fxa found above Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute...
14) Consider the parallelepiped D determined by the vectors (2,-1,2), (1,3, 1), and (2,-1,1). Let T(z, y, 2)a-ytz. Consider the integral I - JSsD TdV. Using the Change of Variables Theorem, write I as an integral of the form T(r(r, s,t), v(r, s, t), z(r, s,t))lJ(r,s, t) dr ds dt for a suitable linear change of variables (r, s, t) (, y,z). The Jacobian J(r,s,t) you get here should be a constant function. 14) Consider the parallelepiped D determined by...
Part D,E,F,G 10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
Let p(x) = 24 + 23 +1€ Z2[2] and let a = [z] in the field E = Z2[z]/(p(x)), so a is a root of p(x). (a) (15 points) Write the following elements of E in the form aa+ba+ca+d, with a,b,c,d € Z2. i. a“, a, a6, and a 10 ii. a5 +a+ + a2 + 1 iii. (a? + 1)4 (b) (5 points) The set of units E* = E-{0} of the field E is a group of order...
- Let X, 1.2.4. U(-1,1) for i € {1, 2, 3). (a) Write down the expected value and variance of X]. Sketch the pdf fxı. [3] (b) Let Y = X1 + X2. Compute the pdf fy of Y and sketch it. Using fy, or otherwise, compute the expected value and variance of Y. (7) (c) Let Z = X1 + X2 + X3 = Y + X3. Using exactly the same technique that you used in part (b), it...
Where And Exercise 6.5.28 Let S (z, y, z) e R3 1 z? + уг + z2-1,#2 0} be the upper hemisphere of the unit sphere in R3. For each of the following integrals, first predict what the integral will be, and then do the computation to verify your prediction 22. 222. 1U. JS Definition 6.5.9 Let S,T C(RT, R). The wedge product of S and T is the alternating bilinear form SAT : Rn × Rn → R given...
Exercise 5.12. Let n∈N and S={0,1,...,n−1}, and suppose that P({s}) = 1/n for each s∈S. Let X:S→R be the random variable defined by X(s) =s. i) Find a closed form formula for MX(z), which does not use sigma notation or any other iterative notation. (ii) Calculate E[X]. (iii) Calculate Var[X].